Find the closed-form solution for the denoised signal ?f for both the weight matrix–based and the Laplacian-based frameworks. Note that only the expression for S 2 ( f) is different in each framework. Hint: Take the derivative with respect to f and equate to zero.

This problem is about graph signal denoising via regularization. Usually the measurements


are corrupted by noise, and the task of graph signal denoising is to recover the true graph


signal from noisy measurements. Consider a noisy graph signal measurement





where


f is the true graph signal and





n is the noise signal. The goal is to recover original signal





f from noisy measurements





y. In the classical signal processing, discrete-time signals and


digital images are denoised via regularization, where the regularization term usually enforces


smoothness. Similarly, a graph signal denoising task can be formulated as the following


optimization problem:





where S2(f) is 2-Dirichlet form as given by Equation (10.5.16) and ? is